Higher Willmore Energies, Q-Curvatures, and Related Global Geometry Problems

Higher Willmore energies, Q-curvatures, and related global geometry problems.

Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurface ... Loewner-Nirenberg-Yamabe problem. in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834;

University of Auckland, Department of Mathematics

MATRIX February 2019

Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; A conformal energy

It was observed by Blaschke 1929, Willmore 1965 and others that Z H2 dA =: Willmore energy (or bending energy 1920) Σ=surf 3 is invariant under the conformal motions of E (basically 3 3 SO(4, 1) action). These are the local maps ρ : E → E that preserve angles.

This property is linked to applications e.g. solid mechanics including cell membranes, relativity and string theory, and geometric analysis – Willmore conjecture concerns absolute minimisers. Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; The Willmore equation

So the Euler-Lagrange equation (wrt variation of embedding) for the Willmore energy is important:

2 ∆ΣH + 2H(H − K) = 0; K = Gauss curvature. | {z } B = Willmore Invariant

The WI is an extremely interesting invariant for embeddings 3 into Euclidean space E , 2 3 ι :Σ −→ E ,

It is conformally invariant and has linear leading term ∆ΣH.A rare quality! One can show that it is a fundamental conformal curvature quantity. Question: What is the meaning of the Willmore invariant? Are there analogues in higher dimensions? This turns out to be linked to the problem of ﬁnding all conformal hypersurface invariants!

(M, g) a (n + 1)-dimensional Riemannian manifold (M, g). Problem (Y): ∃ a smooth real-valued positive function u on M satisfyingg ¯ := u−2g has scalar curvature Scg¯ = −n(n + 1)η? Set u = ρ−2/(n−1) then problem governed by the Yamabe equation:

h n i n+3 − 4 ∆ + Sc ρ + n (n + 1) ηρ n−1 = 0 η = 1, 0, −1. (1) n − 1 – solved by Yamabe, Trudinger, Aubin, Schoen (1960–1984). Problem (sY): ∃ a smooth real-valued positive function u on M satisfyingg ¯ := u−2g has scalar curvature Scg¯ = −n(n + 1)η? [Set u = ρ−2/(n−1) then] problem governed by 2 Sc S(u) := |du|2 − u ∆ + u = η. (2) n + 1 2n Q: Solutions where u changes sign? If so nature of Z(u)? NB: ∃ example solutions η = 1 – e.g. on sphere .

The metricg ¯ := u−2g is Einstein i.e. Ricg¯ = λg¯ iﬀ there is u > 0 s.t.

trace-free(∇a∇bu + Pabu) = 0. (AE) – almost Einstein eqn

where Ric = (n − 1)P + gJ, J := traceg (P). We drop u > 0 and study. Best to replace u by conformal density of weight 1 σ ∈ Γ(E[1]), where E[2n + 2] =∼ (Λn+1(TM))2. Then (AE) is conformally invariant. AE is overdetermined so study by prolongation. It is equivalent to the closed ﬁrst order system:

∇aσ − µa = 0

∇aµb + Pabσ + ρg ab = 0 b ∇aρ − Pabµ = 0

g is the conformal metric, ⊗n+1g : (Λn+1(TM))2 −→' E[2n + 2].

The above system can be collected into a linear connection – in fact the conformally invariant tractor bundle T and connection ∇T . Given g ∈ c this is given by

g 2 T = E[1] ⊕ T ∗M[1] ⊕ E[−1], E[1] := (Λn+1TM) 2(n+1)

T b ∇a (σ, µb, ρ) = (∇aσ − µa, ∇µb + Pabσ + g abρ, ∇aρ − Pabµ ), and ∇T preserves a conformally invariant tractor metric h

b T 3 V = (σ, µb, ρ) 7→ 2σρ + µbµ = h(V , V ).

There is also a 2nd order conformally invariant Thomas operator:

 (n + 2w − 1)wf  g Γ(E[w]) ∈ f 7→ DAf =  (n + 2w − 1)∇af  −(∆f + wJf )

g where J is trace (Pab), so a number times Sc(g). Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; The scale tractor g If IA = (σ, µa, ρ) is a parallel tractor then µa = ∇aσ, and 1 ρ = − n+1 (∆σ + Jσ). This gives the ﬁrst statement of: Proposition 1 I parallel implies IA = n+1 DAσ. So I 6= 0 ⇒ σ is nonvanishing on −2 an open dense set Mσ6=0. On Mσ6=0, g¯ = σ g is Einstein. −2 1 Conversely if g¯ = σ g is Einstein then I := n+1 Dσ is parallel. ¯ 1 Drop parallel and study the scale tractor I := Dσ := n+1 Dσ: g 2 I 2 := I AI = g ab(∇ σ)(∇ σ) − σ(J + ∆)σ (3) A a b n + 1 (cf. (2)) where g is any metric from c and ∇ its Levi-Civita connection. This is well-deﬁned everywhere [on (M, c, σ)], while where σ is non-zero, it computes 2 Scg¯ I 2 = − Jg¯ = − whereg ¯ = σ−2g. n + 1 n(n + 1) 2 −1 ∴ I gives a generalisation of the scalar curvature (× n(n+1) ).

The singular Yamabe equation is the equation I 2 = 1 . Proposition (G. JGP 2010 )

Let (M, c, I ) be Riemannian of signature with I 2 positive. Then Z(σ), if not empty, is a smooth embedded separating hypersurface and the manifold M stratiﬁes into M = M− ∪ M0 ∪ M+ according to the strict sign of σ. The components M \ M∓ are conformal compactiﬁcations of M±.

Proof. 2 g ab 2 From I = g (∇aσ)(∇bσ) − n+1 σ(J + ∆)σ: Along Z(σ) we have 2 ab I = g (∇aσ)(∇bσ)(> 0 by assumption). in particular ∇σ is nowhere zero on Z(σ), and so σ is a deﬁning density. Thus Z(σ) (if 6= ∅) is a smoothly embedded hypersurface by the implicit function theorem.

Thus if I 2 > 0 we have:

M− M0 M+

σ = 0

σ < 0 σ > 0

−2 With g± = σ g on M±. Since σ is a defining density for M0 this exactly means that M \ M∓ conformally compactifies (M±, g±). 2 Then M0 has a conformal structure c0 from g|⊗ TM0 . In particular this applies to a singular Yamabe solution I 2 = 1. But then we’ll see there are additional conditions on the (M0, co) embedding (and (M±, g±) are asymptotically hyperbolic.) Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; Conformal compactification and Loewner-Nirenberg

A conformal compactiﬁcation of a complete Riemannian n+1 manifold (M , g+) is a manifold M with boundary ∂M s.t.: −2 • ∃ g on M, with g+ = u g, where • u a deﬁning f’n for ∂M: ∂M = Z(u)& dup 6= 0 ∀p ∈ ∂M.

⇒ canonically a conformal structure on boundary: (∂M, [g|∂M ]). Our question/variant: Given g (or really c = [g]) can we ﬁnd a deﬁning function u ∈ C ∞(M) for Σ = ∂M s.t.

−2 n+1 Sc(u g) = −n(n + 1)? NB: This satisﬁed for H Ball Answer - Yes: Loewner-Nirenberg, Aviles and McOwen, ACF. Smoothly? Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; The obstruction density of ACF Can we solve Sc(u−2g) = −n(n + 1)? formally (i.e. power series) along the boundary? Answer: No - in general can get: Theorem (Andersson, Chru´sciel,& Friedrich) ∞ −2 n+1 ∃u ∈ C (M) s.t. Sc(u g) = −n(n + 1) + u Bn. and Bn|∂M is the obstrn to smooth soln of L-N. problem. Further

B2|∂M = δ · δ · ˚L + lower order is a conformal invariant of Σ2 = ∂M.

In fact B2 = Willmore Invariant! Theorem.[G. + Waldron 2013] For n ≥ 2 s.t.: •Bn is a conformal invariant of Σ = ∂M. •Bn generalises B2 = B. Idea. Use: Lemma 2 Sc(g+) = −n(n + 1) ⇔ I := h(I , I ) = 1

Problem: For a conformal manifold (M, c) and an embedding ι :Σ → M solve 1 I I A = (D¯ σ)(D¯ Aσ) = 1 + O(σ`), D¯ := D A A n + 1 for ` as high as possible, and σ a Σ deﬁning density. Key point: Proposition (G.+Waldron 2014) 1 A x := σ, y := − I 2 I DA generate an sl(2)! More generally:

[I · D, σ] = I 2(n + 1 + 2w) w = weight operator

From standard sl(2) identities we have [I · D, σk+1] = I 2σk (k + 1)(n + k + 1 + 2w), and this allows an inductive solution (using also other tractor identities).

2 k Iσ = 1 + σ Ak where Ak ∈ Γ(E[−k])

is smooth on M, and k ≥ 1, then 0 k+1 • if k 6= (n + 1) then ∃ fk ∈ Γ(E[−k]) s.t. σ := σ + σ fk 2 k+1 satisﬁes Iσ0 = 1 + σ Ak+1, where Ak+1 smooth; 2 2 n+2 • if k = (n + 1) then: Iσ0 = Iσ + O(σ ).

Proof. Squaring with the tractor metric, using the sl(2), etc

0 2 k+1 2 (D¯ σ ) = (D¯ σ + D¯ (σ fk )) 2 = I 2 + I · D(σk+1f ) + (D¯ (σk+1f ))2 σ n + 1 σ k k 2σk = 1 + σk A + (k + 1)(n + 1 − k)f + O(σk+1). k n + 1 k

Theorem (G.-, Waldron arXiv:1506.02723) For Σn embedded in (Mn+1, c) there is a distinguished deﬁning density σ¯, unique modulo +O(σn+2), s.t. 2 n+1 Iσ¯ = 1 +σ ¯ Bσ¯. Moreover: B := Bσ¯|Σ ∈ Γ(EΣ[−n − 1]) is determined by (M, c, Σ) and is a natural conformal invariant. For n even B = 0 generalises the Willmore equation in that: n B = ∆¯ 2 H + lower order terms; while for n odd B has no linear leading term.

Corollary (ACF + above implies) On a closed (M, g) if there is a sign changing smooth solution of 2 2 Sc sing. Yamabe: |du| − n+1 u ∆ + 2n u = 1 then Σ := Z(u) is a higher Willmore hypersurface – i.e. it satisﬁes B = 0.

For suitable regularisations M of conformally compact manifolds M: Z √ vn v1 Vol = g+ = n + ··· + + A log + Vren + O(). M

Theorem (Graham 2016: PAMS 2017, arXiv:1606.00069) −2 If g+ =σ ¯ g, where σ¯ an approximate solution of the sing. Yamabe problem then A a conformal invariant of Σ ,→ M and

δA (n + 1)(n − 1) = B δΣ 2 So the anomaly term in the renormalised volume expansion provides an energy with functional gradient the obstruction density, in other words an energy generalising the Willmore energy.

In fact – also in analogy with the treatment of Poincar´e-Einstein manifolds – there is nice local quantity giving the anomaly: Theorem (G.- Waldron, CMP 2017, arXiv:1603.07367) 1 Z A = Q n!(n − 1)! Σ

where, with τ ∈ ΓE+[1] a scale giving the boundary metric, Q := (−I · D)n log τ.

• Q here is an extrinsically coupled Q-curvature meaning e.g.

gΣ −nf gΣ 2f Q b = e (Q + Pnf ) where gbΣ = e gΣ and for n even n 2 Pn = ∆Σ + lower order terms; Pn FSA, and Pn1 = 0,

is an extrinsically coupled GJMS type operator. Q and Pn are from G.-, Waldron arXiv:1104.2991 = Indiana U.M.J. 2014. Rod Gover. background: G-. and Andrew Waldron: Conformal hypersurfaceHigher Willmore ... Loewner-Nirenberg-Yamabe energies, Q-curv. and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; Idea of proof

Use a Heaviside function θ to “cut oﬀ” an integral over all M

Z dV gτ σ Vol = n+1 θ( − ). M σ τ Then the divergent terms and anomaly are given by

n−k d n+1 d vk ∼ Vol , dn−k d =0

So Z n−k Z δ (σ) n−1 vk ∼ k and A ∼ δ (σ)I · D log τ M τ M Then via identities, and the sl(2) again Z Z n−k 1 n vk ∼ (I · D) k and A ∼ (I · D) log τ Σ τ Σ

n+1 Conformal compactiﬁcation of H by symmetry breaking:

σ˜=0 n n+1 σ˜=1 S = ∂H

N+ n+2,1 R P+ n+1 H A σ˜ = IAX

I n+3 • Aﬃne parallel transport on R gives the conformal tractor n+1 n+2,1 2 connection on (S , c). Thus spacelike I in R (with I = 1) gives parallel tractor I and hence a solution of singular Yamabe I 2 = 1 s.t. Z(σ) 6= ∅. In this case Σ := Z(σ) is totally umbilic (and higher Willmore). There are solutions on other closed mﬂds: G+Leitner: Commun. Contemp. Math. 12 (2010)

The Obata Theorem states (more than):

Theorem (Obata ) If g¯ is a metric on the round sphere Sn that is conformal to the standard metric g and has constant scalar curvature, then g¯ is Einstein. NB: The metric here takes the formg ¯ = σ−2g where σ is a smooth nowhere vanishing function, and the condition that it has constant scalar curvature is that the scale tractor satisﬁes 2 ¯ Iσ = constant (where recall I = Dσ). Thus related to the Obata Theorem there is a very nice more general question: Question Let (Sn, g) be the usual round sphere. Let σ ∈ C ∞(Sn), possibly 2 −2 with Z(σ) non-empty, such that Iσ = constant. Then is g¯ = σ g necessarily Einstein on Sn \Z(σ)?

Above question is especially interesting if I 2 = const. > 0. Then as a special case of the earlier general result (Corollary): Corollary Let (Sn, g) be the usual round sphere. Let σ ∈ C ∞(Sn), with 2 Z(σ) non-empty, such that Iσ = const. > 0. Then Z(σ) is a (higher) Willmore hypersurface.

2 Above we saw that ∃ solutions to Iσ = const. 6= 0 with Z(σ) totally umbilic. Ifg ¯ = σ−2g is Einstein then Z(σ) umbilic is forced. Hence weaker question: Question Let (Sn, g) be the usual round sphere. Let σ ∈ C ∞(Sn), possibly 2 with Z(σ) non-empty, such that Iσ = constant > 0. Then is Z(σ) necessarily totally umbilic?

Higher Willmore energies, Q-curvatures, and related global geometry problems. The Willmore energy and its functional gradient (under variations of embedding) have recently been the subject of recent interest in both geometric analysis and physics, in part because of their link to conformal geometry. Considering a singular Yamabe problem on manifolds with boundary shows that these these surface invariants are the lowest dimensional examples in a family of conformal invariants for hypersurfaces in any dimension. The same construction and variational considerations shows that (on even dimensional hypersurfaces) the higher Willmore energy and its functional gradient are analogues of the integral of the celebrated Q-curvature conformal invariant and its function gradient (now with respect to metric variations) which is known as the Feﬀerman-Graham obstruction tensor (or the Bach tensor in dimension 4). In fact the link is deeper than this in that the Willmore energy we consider is an integral of an invariant that

Rod Govergeneralises. background: the G-. and Branson Andrew Waldron: Q-curvature.Conformal hypersurfaceHigher This Willmore ... is Loewner-Nirenberg-Yamabe partenergies, of Q-curv. fascinating and global problem. problems in press CAG, arXiv:1506.02723, and Renormalized Volume, CMP, 2017; G-., Waldon, Commun. Contemp. Math., in press, arXiv:1611.0834; unifying picture that includes some interesting open problems in global geometry. This is joint work with Andrew Waldron.